Demos: 1S-02 Compound Pendulum

The pendulum oscillates about the axis X-X in a plane perpendicular to X-X. The weight W is fixed, whereas the weight A can slide up and down the shaft, altering the rotational inertia. The period of the pendulum about O is found from:

where d is the distance to the center of gravity of A and W (assuming the other masses to be negligible). If A and W are also approximated as point masses, then

The center of gravity is located, relative to O,

This leads to

Note that if MA is zero, we have the result for a simple pendulum. If MA = MW and dA = dW, the period of the pendulum is indefinitely large, as expected.

If the entire mechanism is tilted about the hinge through an angle q, the factor g becomes gcosq and the period will increase (Machs Pendulum).

Directions: For a qualitative approach to the demonstration, simply move the sliding mass to various positions and show the positional dependence. In particular, if the mass A is placed such that it is located at the center of gravity, the period will become indefinitely large (infinite). If you wish to perform a quantitative analysis, careful measurements must be made of the distances. (You could measure g if all other parameters are known.)

For the Mach Pendulum, leave the masses in position and lift the apparatus to some intermediate angle to show that the period diminishes with q.

Suggestions for Presentation: See Directions above.

Applications: A variant of the Mach Pendulum was used in astronaut training years ago. The astronaut trainee was attached by harness to a long rope. He moved about on a ramp inclined at an angle such that the component of g along the slope was 1/6 that of the Earth. In some ways, the astronaut was able to simulate movement on the moon.